Why is $\mathsf{D}_{qc}(X)$ the right notion, instead of $\mathsf{D}(\mathsf{QCoh}(X))$?

Question

Let $\mathsf{A}$ be an abelian category and $\mathsf{B}$ be a full abelian subcategory. More often than not, instead of being interested in the derived category $\mathsf{D}(\mathsf{B})$, we are interested in the full subcategory $\mathsf{D}_{\mathsf{B}}(\mathsf{A})$ of $\mathsf{D}(\mathsf{A})$ composed of the complexes whose cohomology is in $\mathsf{B}$. (The cases of quasi-coherent / coherent sheaves and of holonomic $\mathcal{D}$-modules come to mind.)

This seems counter-intuitive for me since we usually have more tools for dealing with objects in the category $\mathsf{B}$ which could be applied to $\mathsf{D}(\mathsf{B})$. While both categories are usually equivalent, I feel that people usually think that $\mathsf{D}_{\mathsf{B}}(\mathsf{A})$ is the right object. Why is it so?

Answer 1

The triangulated category $\mathsf{D}_{\mathsf{B}}(\mathsf{A})$ can be promoted to a stable $\infty$-category. One of the many interests of working with stable $\infty$-categories is that we have a reasonable theory of descent for them: we can define sheaves of stable $\infty$-categories (it is possible to formulate such concepts using the language of dg categories). However, the assignment $$X\mapsto \mathsf{D}(\mathsf{QCoh}(X))$$ is unfortunately not a sheaf of stable $\infty$-categories on the big site of all schemes (for the Zariski or fppf topology, say) and its sheafification leads to the assignment $$X\mapsto \mathsf{D}_{qc}(X)\, .$$ We have $\mathsf{D}(\mathsf{QCoh}(X))\cong \mathsf{D}_{qc}(X)$ locally (in fact for $X$ any separated and quasi-compact scheme), so that $\mathsf{D}_{qc}(X)$ gets many of the nice features of $\mathsf{D}(\mathsf{QCoh}(X))$ by local-to-global principles.